Analytic descriptions of ultracold electron-atom collisions

From the quantum defect theory\footnote{Gao, PRL, \textbf{104}, 213201 (2010); PRA, \textbf{88}, 022701 (2013).} (QDT) and the multichannel quantum defect theory\footnote{Li \textit{et al.}, PRA \textbf{89}, 052704 (2014); Li and Gao, PRA \textbf{91}, 032702 (2015).} (MQDT) for an attractive polarization potential, we derive both a QDT expansion and a MQDT expansion, that together provide analytic descriptions of low-energy electron collisions with atoms in a ground $^1S$ or a ground $^2S$ state. The expansions are accurate over an energy range from the zero kelvin to hundreds of kelvins, and include effects of hyperfine structure if it is present in cases of $^2S$ atoms. Results for electron-hydrogen hyperfine-changing collisions are presented as an example.